Growth Rate
Johannes Kepler discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence {Fn} approaches the golden ratio which is approximately 1.61803. In 1765, Leonhard Euler published an explicit formula, known today as the Binet formula,
It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio φ.
In 1960, Hillel Furstenberg and Harry Kesten showed that for a general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the nth root of |fn| converges to a constant value almost surely, or with probability one:
An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating point arithmetics validated by an analysis of the rounding error.
Read more about this topic: Random Fibonacci Sequence
Famous quotes containing the words growth and/or rate:
“Of all the wastes of human ignorance perhaps the most extravagant and costly to human growth has been the waste of the distinctive powers of womanhood after the child-bearing age.”
—Anna Garlin Spencer (1851–1931)
“Strange that the vanity which accompanies beauty—excusable, perhaps, when there is such great beauty, or at any rate understandable—should persist after the beauty was gone.”
—Mary A. [Elizabeth, Countess Von] Arnim (1866–1941)